Yes. Oracle is enterprise grade software.
I'm not sure if its really any more stable that mysql, I haven't used mysql that much, but I dont ever remember having mysql crash on me. I've had oracle crash, but when it does, it gives me more information about why it crashed than I could possibly want, and Oracle support is always there to help ( for a fee ).
Its very very robust, Oracle DB will do virtually everything it can before breaking your data, I've had mysql servers do really weird things when they run out of disk space, Oracle will just halt all transactions, and eventually shutdown if it can't write the files it needs. I've never lost data in oracle, even when I do stupid things like forget the where clause and update every row rather than a single row, its very easy to get the database back to how it was before screwing up.
Not sure about security, certainly Oracle gives you lots of options for how you are going to connect to the DB and authenticate. It gives lots of options regarding which users have access to what, etc. But as with most things, if you want to take security seriously, then you need an expert to do it. Oracle certainly has a lot more to lose if they don't get security right. But, as with all things there has been exploits.
If nothing else, just consider this... When Oracle stuffs up, they have customers who are paying $40k per CPU (if they are suckers and pay list price) license + yearly maintenance fees.. This gives them a very strong intensive to make sure the customers are happy with the product.
For a small database, I'd seriously recommend Oracle XE well before mysql. It has the important features of mysql (Free), its dead easy to install, comes with a nice web interface and application framework (Application Express), if you DB will happy run on a single cpu, 1gb ram and 4gb data, then XE is the way to go IMHO.
Mysql has its uses, many many people have shown that you can build great things with it, but its far behind oracle (and SQL Server, and DB2) in terms of features... But then, its also free and very easy to learn, which for many people is the most important feature.